I wonder what are the "moral" differences between Poincare and Sobolev inequalities. Let me state them (hopefully without errors) using Wikipedia as source:
Poincare inequalities Let $\Omega$ be a domain in $\mathbb R^n$ bounded in at least one direction, then there exists some $C = C(p, \Omega)$ such that for all $u \in W^{1,p}_0(\Omega)$ ($1\leq p < \infty$) $$ \Vert u \Vert_{L^p(\Omega)} \leq C \Vert \nabla u \Vert_{L^p(\Omega)} $$ We can also change the condition of $u$ belonging in $W^{1,p}_0(\Omega)$ by $u \in W^{1,p}(\Omega)$ and $\int_\Omega u dx = 0$ (only for bounded $\Omega$). This is called Poincare-Wirtinger.
Gagliardo-Nirenberg-Sobolev inequality Assume $u \in W^{1,p}(\mathbb R^n)$ (in Wikipedia they say $u\in C^1(\mathbb R^n)$ with compact support but by density I think it doesn't matter, correct me if I'm wrong please). Then for $p \in [1,n)$, there exists some $C=C(p)>0$ such that $$ \Vert u \Vert_{p^*} \leq C \Vert \nabla u \Vert_{p} $$ where $p^*$ is some real number larger than $p$ and depending on $n$ and $p$.
I can see that Poincare inequalities are intimately related to domains, whereas GNS isn't. But at first glance, GNS is stronger because if we assume $\Omega$ bounded then $u \in W^{1,p}_0(\Omega) \subset W^{1,p}(\mathbb R^n)$ (by extending $u$ by $0$ outside of $\Omega$) and moreover $\Vert \nabla u \Vert_p \lesssim \Vert \nabla u \Vert_{p^*}$ by Holder ineq.
Summing up, it seems to me that in GNS we don't ask any condition on $u$ besides being in $W^{1,p}$ and we get an (intuitively) better bound on its $p$-norm because it "uses" lower integrability on $\nabla u$.
A related very vague question, whenever I am working in a domain, when should I be thinking about Poincare inequality and when about GNS inequality?